The critical factor is the current through the primer in the flash bulb! With three "D" cells four bulbs wired in parallel should work well however since you are using "C" cells there may not be enough current available to fire four bulbs. The voltage only matters in there being enough to push the needed current through the resistance of the primers. While I do not know what the minimum voltage to fire a flash bulb is, I do know that 3 volts with a sufficient current available will reliably fire one.
If you wire the bulbs in series you need 3n volts (where "n" is the number of bulbs) or 12 volts for four bulbs. your three "C" cells (or three "D" cells) only have a total voltage of 4.5 volts. The "D" cells have the same voltage as "C" cells but have more current available before the internal resistance of the cells causes an unacceptable voltage drop.
If the bulbs are wired in parallel there will be the total voltage across each bulb and the current available from the batteries will be divided between the bulbs.
A "B-C pack" has a much greater momentary current available than a battery so that the voltage to which the capacitor is charged governs how many bulbs wired in series can be fired with it.

My plan... tape a 22,000uf/10v cap to the gun and feed it with a series isolation diode and current limiting resistor connected to the focuspot terminals for charging purposes. From the cap feed the bulbs in parallel with an extension cord connected in series to the shutter sync terminals. I'll be using four clamp on Edison base reflectors and No.2 bulbs plugged into a household line strip. At 20' that should give me f32 with Plus-X... perfect exposure.

On the other hand maybe I should just use a 6 or 12V lantern battery in a shoulder bag and be done with it.